Limit Time Optimal Synthesis for a Control-Affine System on S²

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For α∈(0,π/2), let (Σ)α be the control system x=(F+uG)x, where x belongs to the two-dimensional unit sphere S², u∈ [-1,1] and F,G are 3×3 skew-symmetric matrices generating rotations with perpendicular axes of respective length (α) and (α). In this paper, we study the time optimal synthesis (TOS) from the north pole (0,0,1)ᵗ associated to (Σ)α, as the parameter α tends to zero. We first prove that the TOS is characterized by a ``two-snakes'' configuration on the whole S², except for a neighborhood Uα of the south pole (0,0,-1)ᵗ of diameter at most Ø(α). We next show that, inside Uα, the TOS depends on the relationship between r(α):=π/2α-[π/2α] andα. More precisely, we characterize three main relationships, by considering sequences (αₖ)ₖ≥ ₀ satisfying (a)r(αₖ)=r; (b) r(αₖ)=Cαₖ and (c) r(αₖ)=0, where r∈ (0,1) and C>0. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as α tends to zero, of the corresponding TOS inside Uα.